We present a new characterization of k-trees based on their reduced clique graphs and \((k+1)\) -line graphs, which are block graphs. We explore structural properties of these two classes, showing that the number of clique trees of a k-tree G equals the number of spanning trees of the \((k+1)\) -line graph of G. This relationship allow us to present a new approach for determining the number of spanning trees of any connected block graph and to address a special case of a “reverse problem" raised in the literature. We show that these results can be accomplished in linear time complexity.
